Could you share a link to your site?
Any webhost should do the trick, so the main differentiator would be your budget. If your site is new, it should barely draw ressources so either go for shared or semi-dedicaced (classic shared with a higher ressource allocation. typically over twice higher). Make sure you use some caching first if available, and maybe public CDNs for libraries like jquery.
Elkate appears to be compatible with php7 (since 1.1 beta1 ?), so going for a webhost with php7 available should somewhat reduce your ressource consumpton usage and make a better usage of the limited ressources allocated, when the stable branch of Elkarte is eventually fully compatible.
Thanks for the suggestion, my host said the 1 time spike was due to some unknown instance. If it happens again they will provide me with a detailed report. So staying with my host to get it fixed first. And i will surely take your advice on php7
It's indeed an odd spike, according to the info give here, it looks like a bot with the Google Page speed Insight user agent, that hits about 10 pages at the very same moment.
Having the host a limit on 10 "entry processes", probably the server hits the limit and reports the exceeding of the limit.
I suggested to block the bot by user agent at the htaccess level.
Also because the IPs look legit (i.e. from the google range), so I wonder if it is some new aggressive random bot testing stuff like that via API or something.
I think your error was related to your hosting, until last year I hosted 3 forums on a cheap shared hosting and never seen a similar error (1 smf forum with 25.000 messages, 1 elkarte forum with about 50.000 messages at that time and 1 XF forum)
If you were to continue your project, as you're already reching limits on your shared hosting, I'd look for a host with higher limits (Most will be flexible with telling you these) or look at migrating the project to a managed VPS with more dedicated resources.